Jacob had 50% more pencils than Gavin. Joseph had 20% fewer pencils than Jacob. Jacob and Gavin gave Joseph some pencils in the ratio of 1 : 3. As a result, Joseph had 50% more pencils than before. Given that Joseph had 50 more pencils than Gavin in the end, how many pencils did Jacob give to Joseph?

Jacob	Gavin	Joseph
3x5	2x5
5x3		4x3
15	10	12

100% + 50% = 150%
150% = ¹⁵⁰₁₀₀ = ³₂

Jacob : Gavin = 3 : 2

100% - 20% = 80%
80% = ⁸⁰₁₀₀ = ⁴₅

Jacob : Joseph = 5 : 4

The number of pencils that Jacob had at first is repeated. Make the number of pencils that Jacob had at first the same.  LCM of 3 and 5 is 15.

	Jacob	Gavin	Joseph
Before	15x2 =  30 u	10x2 = 20 u	12x2 = 24 u
Change			+ 6x2 = + 12 u
Comparing the changes in the number of pencils	- 1x3 = - 3 u	- 3x3 = - 9 u	+ 4x3 = + 12 u
After	27 u	11 u	18x2 = 36 u

50% x 12 
= ⁵⁰₁₀₀ x 12 
= 6 

The total number of pencils that Jacob and Gavin gave to Joseph is repeated.  Make the total number of pencils that Jacob and Gavin gave to Joseph the same.  LCM of 4 and 6 is 12.

Number of pencils that Joseph had more than Gavin 
= 36 u - 11 u
= 25 u

25 u = 50

1 u = 50 ÷ 25 = 2

Number of pencils that Jacob gave to Joseph
= 3 u
= 3 x 2
= 6

Answer(s): 6